Graphics Reference
In-Depth Information
Figure 12.4.
Interesting surfaces of revolution.
Since p(1,p/4) =
p
, it follows that
∂
p
x
∂
p
)
=
(
)
(
)
(
(
)
=-
14 11212
,
p
,
,
and
14 0 22
,
p
,
,
∂
∂q
are a basis for our tangent plane. Finally,
∂
∂
p
∂
∂q
p
(
)
(
)
¥
(
)
=- -
14
,
p
14 2 2 2
,
p
,
,
x
is a normal vector to the plane, so that its equation is
(
)
∑
(
(
)
-
(
)
)
=
22 2
,
--
,
xyz
, ,
122 0
,
,
,
or
22220
x
-
y
-
z
+
=
.
12.2.2 Example.
Assume that
S
is the surface of revolution obtained by rotating
the segment
X
= [(3,1),(3,3)] about the x-axis. We want to find the tangent plane to
S
at
p
= (3,0,2).
Solution.
The surface
S
is parameterized by
(
)
=
(
)
py
,
q
3
,
y
cos ,
q
y
sin
q
and
∂
∂
p
y
∂
∂q
p
(
)
=
(
)
(
)
=-
(
)
y
,
q
0
,cos ,sin
q
q
and
y
,
q
0
,
y
sin ,
q
y
cos
q
.